Presentations from Advances in continuum quantum Monte Carlo methods CECAM August2007
From QMC
[edit] Abstract List
[edit] Presentations
[edit] Computing derivatives and small differences of diffusion Monte Carlo energies
by Roland Assaraf, Alexander Kollias and Michel Caffarel
Abstract Many quantities of physical or chemical interest can be computed as differences of groundstate energies. These quantities include for example transition state energies, binding energies, but also any property of the groundstate (derivative of the energy). Quantum Monte Carlo methods (QMC) can be considered today as the benchmark methods for groundstate energies. However exploiting this advantage to obtain small differences is not easy, due to the statistical errors (they are small for the energy but often too large for the differences). Today, we have at our disposal a general solution for this problem, in Variationnal Monte Carlo (VMC), but not in the more accurate diffusion Monte Carlo (DMC) method, unless some uncontroled approximation is introduced. In this talk, I will present a promising unbiased approach based on a generalized correlated sampling scheme, to compute small differences, and derivatives of DMC energies. As a preliminary application, we perform accurate calculations of small differences of surface potential energies for a variety of molecules, and forces on nuclei, the latter beeing obtained exactly as the derivative of the DMC fixed-node energy with respect to the nuclei positions.
[edit] Finite-size correction in many-body electronic structure calculations
by Shiwei Zhang, Hendra Kwee, Henry Krakauer
Abstract Many-body electronic structure calculations in simulation cells with periodic boundary conditions have finite-size errors. As quantum simulation methods become more accurate, eliminating the bias from finite-size errors has become more important and has drawn considerable attention. Reducing the errors using increasingly larger simulation cells is computationally costly. Finite-size corrections from Hartree-Fock or density functional theory (DFT) are computationally affordable and often used, but are in general not accurate. Here we present a correction method that is specifically designed to approximately include two-body finite-size corrections in a modified DFT calculation, via a finite-size exchange correlation functional. The method is simple, and gives post-processing corrections that can be applied to any many-body results. Applications to the P$_2$ molecule, bulk semiconducting silicon, and sodium metal show that the method yields improved estimates of the finite-size errors. In the last part, time permitting, I will give a report on recent developments in the phaseless auxiliary-field Monte Carlo method, including calculations in molecular systems.
[edit] Hellman-Feynman operator sampling in Diffusion Monte Carlo calculations.
by René Gaudoin, Txema Pitarke slides
Abstract Diffusion Monte Carlo (DMC) calculations typically yield highly accurate results in solid-state and quantum-chemical calculations. However, operators that do not commute with the Hamiltonian are at best sampled correctly up to second order in the error of the underlying trial wavefunction, once simple corrections have been applied. Operators that suffer from these problems include potential energies and the density. Based on the Hellman-Feynman theorem we present a method for the correct DMC sampling of all operators diagonal in real space. Our method is memory-efficient and easy to implement in any standard DMC code.
[edit] Diffusion Monte Carlo methods with non-local potentials
by Michele Casula
Abstract We show how to include non-local potentials in the diffusion Monte Carlo framework in a variational way, substantially improving both the accuracy and the computational stability upon previous non-variational diffusion Monte Carlo approaches. We will review and compare two methods, the lattice regularized diffusion Monte Carlo [1] and the non-local diffusion Monte Carlo [2], which are variational even in the presence of non-local pseudopotentials. These methods can open the route for even more reliable and accurate electronic structure calculations of the ground state by means of quantum Monte Carlo.
[1] M. Casula, C. Filippi, and S. Sorella, Phys. Rev. Lett. 95, 100201 (2005). [2] M. Casula, Phys. Rev. B 74, 161102(R) (2006).
[edit] The Fixed phase Method and some properties of complex wave functions
by David Ceperley, Jaron Krogel slides
Abstract The fixed phase method is the generalization of the fixed-node method for complex trial functions. While complex wavefunctions are slower to implement than real wavefunctions, they possess some advantages, e. g. to treat finite size effects. As is known for real functions, the "nodal action" and hence the "phase action" can be an important contributor to finite size time step bias. To understand how to construct good phase action, we have studied the phase of many-body mean field wave functions, both for insulators and for metals. We have found that the gradient of the phase becomes increasingly steep as the number of particles increases. Roughly speaking the "twisted" boundary conditions become less important as the system size grows, so that a complex wavefunction approaches a real wavefunction. Hence, for large systems, actions for real wavefunctions become relevant to use with complex wavefunctions.
[edit] Energy-consistent pseudopotentials for quantum Monte Carlo calculations
by Mark Burkatzki, C.Filippi, M. Dolg
Abstract We present scalar-relativistic energy-consistent Hartree-Fock pseudopotentials for the main-group [1] and 3d-transition-metal [2] elements. The pseudopotentials do not exhibit a singularity at the nucleus and are therefore suitable for quantum Monte Carlo (QMC) calculations. We demonstrate their transferability through extensive benchmark calculations of atomic excitation spectra as well as molecular properties. In particular, we compute the vibrational frequencies and binding energies of 26 first- and second-row diatomic molecules using post Hartree-Fock methods, finding excellent agreement with the corresponding all-electron values. We also show that our pseudopotentials give superior accuracy than other existing pseudopotentials constructed specifically for QMC. We will also present preliminary QMC calculations for selected atomic and diatomic 3d-transition-metal systems. Finally, valence basis sets of different sizes (VnZ with n=D,T,Q,5 for 1st and 2nd row; with n=D,T for 3rd to 5th row; with n=D,T,Q for the 3d transition metals) optimized for our pseudopotentials are also presented.
[1] M. Burkatzki, C. Filippi, M. Dolg, J. Chem. Phys. 126, 234105 (2007). [2] M. Burkatzki, C. Filippi, M. Dolg, in preparation.
[edit] Quantum Monte Carlo for some difficult systems of Quantum Chemistry
by Michel Caffarel, A. Ramirez-Solis
Abstract In this talk I present some recent appplications of QMC to several systems difficult to describe using standard methods of computational quantum chemistry. Applications include molecular systems containing transition metal atoms and polyoxygen species.
Key references 1. M. Caffarel, J.P. Daudey, J.L. Heully, and A. Ramirez-Solis, "Towards accurate all-electron quantum Monte Carlo calculations of transition metal systems: Spectroscopy of the copper atom", J. Chem. Phys. vol. 123, 094102 (2005).
2. A. Ramirez-Solis and M. Caffarel, "The application of quantum Monte Carlo to the spectroscopy of metallic molecules", Recent Res. Develop. Chem. Phys. Editor R. Hernandez-Lamoneda. Transworld Research. Kerala, India (2007).
3. M. Caffarel, R. Hernandez-Lamoneda, A. Scemama, and A. Ramirez-Solis "On the O4 debate: a multireference quantum Monte Carlo study" (submitted).
[edit] Projection Monte Carlo: Branching vs. Metropolis
by Saverio Moroni slides
Abstract In a "linear scaling" algorithm, the cost of calculating the (exact or fixed-node) ground-state energy per particle with a given statistical uncertainty is independent of the system size. The computational efficiency as a function of the number of particles is studied for two variants of Projection Monte Carlo, namely Diffusion Monte Carlo (a branching random walk) and Variational Path Integral (a Metropolis random walk). Simulations of bulk liquid 4He with systems of 64 to 512 atoms indicate that linear scaling can be achieved with VPI but not with DMC.
[edit] Progress on Fermion QMC based on path resummations in Slater determinant space
by Ali Alavi
Abstract Our group has, for a number of years, been developing an approach to Fermion QMC which is based on the idea of sampling "graphs", rather than "paths". We will present some recent progress with this general idea, concentrating a new resummation formula which enables us to sum over all paths which visit double (and single) excitations of a reference determinant (eg Hartree-Fock or Kohn-Sham). We show that this approximation produces remarkably similar results to the CCSD method for systems as wide ranging as the N2 molecule and the Ne dimer. What is remarkable about this approximation is its simplicity, requiring a computational effort similar to MP2 (though with heavier memory requirements). We have also implemented this technique for periodic systems using Kohn-Sham orbitals, and we will present energy-volume curves for some simple crystals. If time permits, we will also discuss an extention of the idea of graph sampling to the perturbation series, such as MPn, which allows us to sample the n-th order diagrams, rather than the conventional "sum over all of them". We demonstrate that a simple stochastic algorithm can produce an improved scaling with number of electrons compared to the conventional method.
[edit] Resonating Valence Bond wave function for high pressure hydrogen
by Sandro Sorella, C. Attaccalite | Slides, 11MB
Abstract We introduce a novel technique for ab-initio molecular dynamics simulation of electronic systems within the Born-Oppenheimer approximation. We compute explicitly the forces acting on the ions by solving the Schroedinger equation for the electronic ground state with a variational approach based on a convenient representation of the Resonating Valence Bond wave function[1] and the Quantum Monte Carlo technique (QMC). We also use a new second order Langevin dynamics containing a particular friction matrix that is determined in a consistent way during the simulation. Indeed it can be shown that the non-diagonal part of this matrix allows to correct for the statistical correlation between the force components obtained by QMC. In this way it is possible to simulate finite temperature electronic systems with very high efficiency, while the variational parameters are consistently optimized during the ionic dynamics. We apply this method to clarify the still controversial low-temperature and high-pressure phase diagram of Hydrogen. We find a remarkable stability of the liquid phase in the low temperature high pressure regime, that can be understood within the Resonating Valence Bond picture[2].
[1] P. W. Anderson Science 235, 1196 (1987). [2] S. Sorella and C. Attaccalite, cond-mat/0703800.
[edit] QMC Studies of Quantum Nanowires and Nanolayers
by Neil Drummond, Richard Needs
Abstract Quantum Monte Carlo (QMC) methods have been used to obtain accurate binding-energy data for pairs of parallel, thin, metallic wires and layers modelled by 1D and 2D homogeneous electron gases. We compare our QMC binding energies with results obtained within the random phase approximation, finding significant quantitative differences and disagreement over the asymptotic behaviour for bilayers at low densities. We have calculated pair-correlation functions for metallic biwire and bilayer systems. Our QMC data could be used to investigate van der Waals energy functionals.
[edit] How reactive are electron pairs in molecules? Where are they localized? A QMC/EPLF(n) study.
by Alan Aspuru-Guzik, Carlos Amador-Bedolla, Romelia Salomon-Ferrer, William A. Lester, Jr.
Abstract Electrophilic amination is an appealing synthetic strategy to construct carbon-nitrogen bonds. The authors explore the use of the quantum Monte Carlo method and a proposed variant of the electron pair localization function—the electron pair localization function density—as a measure of the nucleophilicity of nitrogen lone pairs as a possible screening procedure for electrophilic reagents.
Key references (1) C. Amador-Bedolla, R. Salomón-Ferrer, W. A. Lester, Jr, J. A. Vázquez-Martínez, and A. Aspuru-Guzik J. Chem. Phys. 126, 204308 (2007). (2) A. Scemama, P. Chaquin, and M. Caffarel, J. Chem. Phys. 121, 1725 (2004).
[edit] Optimization of quantum Monte Carlo wave functions and calculation of pair densities
by Julien Toulouse, Cyrus J. Umrigar, Roland Assaraf, Claudia Filippi, Sandro Sorella, Richard G. Hennig Talk slides
Abstract We present a simple, robust and highly efficient method for optimizing the parameters of many-body wave functions by energy minimization in quantum Monte Carlo calculations. Using a strong zero-variance principle, the optimal parameters are determined by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its derivatives [1-2]. We apply this method to obtain accurate multideterminant Jastrow-Slater wave functions for atomic and molecular systems, where the Jastrow parameters, the configuration state function coefficients, the orbital coefficients and the basis function exponents are simultaneously optimized. This allows us to reach near chemical accuracy on the dissociation energies of the first-row diatomic homonuclear molecules [3]. We also illustrate the quality of the obtained wave functions by calculating accurate spherically and system-averaged pair densities using improved statistical estimators [4].
Key references [1] J. Toulouse and C. J. Umrigar, J. Chem. Phys. 126, 084102 (2007).
[2] C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig, Phys. Rev. Lett. 98, 110201 (2007).
[3] J. Toulouse, C. J. Umrigar, in preparation.
[4] J. Toulouse, R. Assaraf, C. J. Umrigar, J. Chem. Phys. 126, 244112 (2007).
[edit] Optimization on random surfaces
by John Trail, R.J. Needs
Abstract Monte Carlo methods provide a well established approach to solving the many body Schroedinger equation for large systems. The cost of this approach is the introduction of a random error in estimated quantities, such as the total energy of a system, and this error is generally assumed to be Normally distributed. Here we analyse the true statistics of estimates, and in particular the estimated surfaces whose minima are sought in the application of `Variational Monte Carlo' methods.
[edit] Topologies of fermion nodes: implications for quantum many-body
by Lubos Mitas
Abstract The talk will be focused on implications of the topologies of fermion nodes and nodal cells on properties of many-body wavefunctions and on physical effects. Building upon our previous work on proofs of two nodal cells for noninteracting systems and BCS wavefunctions, I will talk about the proof of two nodal cells systems with general interactions, connections to other physical phenomena and about insights into the efficiency of accurate nodes descriptions for variety of known wavefunctions.
[edit] Optimization of nodal hypersurfaces in quantum Monte Carlo
by Arne Luechow
Abstract Over the last few years, quantum Monte Carlo methods have shown to be applicable to a wide range of quantum chemical problems with high accuracy, including excited states and weak interactions. QMC can retain the inherent favorable scaling of Monte Carlo methods in these applications. Recent results are discussed with the emphasis on new ways to control the fixed-node error of diffusion quantum Monte Carlo by optimization of the nodal hypersurfaces of a trial function.
[edit] Linear and non-linear dielectric response of periodic systems from quantum Monte Carlo
by Paolo Umari
Abstract We present a novel approach that allows to calculate the dielectric response of periodic systems in the quantum Monte Carlo formalism. We employ a many-body generalization for the electric enthalpy functional, where the coupling with the field is expressed via the Berry-phase formulation for the macroscopic polarization. A self-consistent local Hamiltonian then determines the ground-state wavefunction, allowing for accurate diffusion quantum Monte Carlo calculations where the polarization's fixed point is estimated from the average on an iterative sequence. The polarization is sampled through forward-walking. This approach has been validated for the case of the polarizability of an isolated hydrogen atom, and then applied to a periodic system. We then calculate the linear susceptibility of molecular-hydrogen chains with different bond-length alternations. The results found are in excellent agreement with the best estimates obtained from the extrapolation of quantum-chemistry calculations. By adopting trial wavefunctions obtained from Hartree-Fock, we can then calculate with great accuracy the second hyper-susceptibility. Finally, we assess the importance of electronic correlations for the calculated linear and non-linear susceptibilities.
[edit] Quantum Monte Carlo Calculations of Point Defects in Alumina
by Nicholas Hine, Matthew Foulkes Talk slides
Abstract The structural, mechanical, catalytic, electronic and optical properties of materials such as Al2O3 are strongly influenced by the presence of and charge states of defects such as oxygen vacancies. Previous electronic structure calculations of formation energies and geometries of native point defects have produced rather contradictory results, indicating that a more accurate treatment of the electronic structure of the defect is required. Additionally, within DFT, untested assumptions are frequently used, relating to chemical potentials of atoms and electrons, correction of eigenvalues of defect levels due to DFT's well known band gap problem, and electrostatic interactions between neighbouring charged defects. These assumptions strongly affecting the relative ordering of different charge states. QMC is the only method able to combine near-“chemical” levels of accuracy with sufficiently favourable scaling with system size to allow calculations on supercells large enough to converge defect properties. Previous DMC calculations on vacancies and interstitials in carbon and silicon have produced significantly different formation energies to corresponding DFT calculations, suggesting DFT is overbinding the solid relative to DMC, resulting in reduced formation energies for interstitials and increased formation energies for vacancies. We have carried out DFT and DMC calculations on Oxygen vacancies and Oxygen interstitials in a range of charge states in Al2O3 to obtain geometries and formation energies. We find further evidence of DFT overbinding, errors due to self-interaction of localized defect states, and inaccuracy of the bandgap correction, suggesting that DFT may in many situations be inadequate for accurate ab initio calculation of defect formation energies.
[edit] Zero temperature calculations of quantum liquids
by Markus Holzmann slides
Abstract I will adress two main issues of improving fermionic quantum liquid calculations via QMC: how to change the nodal structure via different functional forms of the trial wavefunction and how to perfom extrapolations of energies and other observables to the thermodynamic limit.
[edit] High pressure Hydrogen: new predictions by coupled electron-ion Monte Carlo
by Carlo Pierleoni, David Ceperley, Kris Delaney, Markus Holzmann
Abstract The physical behaviour of hydrogen under megabar compression is still largely unknown because experimental difficulties prevent a systematic investigation of the phase diagram in the interesting range of pressures and temperatures. Ab-initio theoretical methods have been exploited to interpret the scattered experimental data and to make predictions. However the problem is particularly difficult because the energy scales of several different phenomena, such as electronic correlations, protonic quantum effects, finite size effects in the metallic phase and in the metal-insulator transition region, becomes comparable and need to be considered together in a non perturbative manner. Quantum Monte Carlo methods are unique in their ability to treat all those effects with high accuracy in simple systems. In recent years we have developed a new QMC method, the Coupled Electron-Ion Monte Carlo (CEIMC) [1], which is particularly suitable to study hydrogen under high pressure. After a brief introduction to the physics of high pressure hydrogen, I will describe the main ingredients of the new method and present results from CEIMC. In particular I will discuss the metallic phase of hydrogen at pressures beyond the molecular dissociation threshold, including the melting of the proton crystal [2], and the molecular dissociation phenomena in the liquid phase [3].
Key references [1] C. Pierleoni and D. M. Ceperley, Lecture Notes in Physics vol 741 pp 641-683 (2006), xxx.lanl.gov/abs/physics/0510254.
[2] C. Pierleoni, D.M. Ceperley and M. Holzmann, Phys. Rev. Lett. 93, 146402 (2004).
[3] K. Delaney, C. Pierleoni and D.M. Ceperley, Phys. Rev. Lett. 97, 235702 (2006).
[edit] Quantum Monte Carlo calculations of geomaterials
by Dario Alfe
[edit] Backflow wave functions in quantum Monte Carlo
by Pablo Lopez-Rios
Abstract The quality of the nodal surface of trial wave functions determines the accuracy of the results obtained within the fixed-node diffusion Monte Carlo method. Backflow transformations are capable of improving the nodal surface of Slater-Jastrow wave functions in homogeneous and inhomogeneous systems. The transformation can be compactly parametrized so that the number of optimizable parameters is essentially independent of system size, and the parametrized functions can be smoothly truncated in space to achieve favourable scaling with system size. In this talk I will give examples of the results and that we have obtained using backflow transformations on Slater-Jastrow wave functions.
[edit] Approach Toward Linear Time Quantum Monte Carlo
by Bryan Clark, David Ceperley
Abstract Quantum Monte Carlo simulations of fermions scale naively as O(n^3) operations. This scaling has generically kept accessible system sizes to 1000 fermions or fewer. The asymptotically dominate operation involves calculating the ratio of two (similar) determinants. We explore a number of methods designed to accelerate calculating these ratios including methods using iterative techniques, sparse approximate inverses, and truncation.
[edit] Correlations and localization in planar quantum dots
by Cyrus Umrigar a, Devrim Guclu b, Amit Ghosal b, Harold Baranger b, Gun-Sang Jeon c and Jainendra Jain c a Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853 b Physics Department, Duke University, Durham, North Carolina 27708 c Physics Department, Pennsylvania State University, University Park, Pennsylvania 16802
Abstract Quantum dots, also known as ”artificial atoms” are not only of considerable technological interest but also of theoretical interest because it is possible to go from a weak correlation to a strong correlation regime either by increasing the relative strength of electron-electron interaction to the external potential or by increasing the magnetic field. We employ diffusion Monte Carlo to study the ground and excited states of dots in various regimes and compare the results to those from the Hartree Fock (HF) method and from density functional theory within the local spin density approximation (LSDA). In the absence of a magnetic field we find, in contrast to the situation for real atoms, that the total energies and addition energies obtained from LSDA are much more accurate than those from HF. This is because the relative magnitude of the correlation energy to the exchange energy is much larger in dots than in atoms and the density is less inhomogeneous in dots. LSDA predicts reasonably accurate excitation energies for many states, but in those cases where the LSDA states are spin contaminated it predicts excitation energies that are too low, whereas, in those cases where there is considerable multideterminantal character in the excited state it predicts excitation energies that are too high. Hund’s first rule is satisfied for all electron numbers studied, but for N=10 there is a near degeneracy. For strongly correlated dots, highly spin-polarized states, that require promoting electrons between non-interacting shells, become nearly degenerate with the Hund’s rule state. In the large magnetic field limit the determinants can be limited to those arising from the lowest Landau level. For finite magnetic fields Landau level mixing is important and can be taken into account very effectively by multiplying the infinite-field determinants by a Jastrow factor which is optimized by variance minimization. We apply these wave functions to study the transition from the maximum density droplet state (integer quantum Hall state, L = N(N−1)/2) to lower density droplet states (L > N(N −1)/2). Composite-fermion wave functions, projected onto the lowest Landau level and multiplied by an optimized Jastrow factor, provide an accurate and efficient alternative form of the wave functions. We argue that Coulomb blockade phenomena are a useful probe of the cross-over to correlation- induced localization in quantum dots. Through calculations at low density (up to rs 55), we find that the addition energy shows a clear progression from features associated with single-particle shell structure to those caused by commensurability of a Wigner crystal. This cross-over is, then, a signature of interaction-driven localization; for spin-polarized electrons, it occurs near rs 20. As the addition energy is directly measurable in Coulomb blockade conductance experiments, this provides a direct probe of localization in the low density electron gas.
[edit] Posters
[edit] Faster QMC for Periodic Systems Using Polynomial-based Orbitals
by William Parker, John Wilkins, Richard Hennig, Cyrus Umrigar
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Abstract Computing the wave function can be the most time-intensive part of a QMC calculation. Orbitals represented by extended basis functions such as plane waves scale in evaluation time as O(N3) while localized basis functions scale as O(N2). Two methods of localizing the orbital representation are: piecewise-polynomial (pp) interpolation and transformation to a localized basis. The Lagrange form of the pp-interpolant is simple but has discontinuous derivatives at sampling points. The pp-spline(1) is continuous in certain low derivatives. The B-spline(2) shares the pp-spline's derivative continuity but is a transformation not an interpolant of the plane waves (does not reproduce the plane wave sum at the sampling points). These methods have O(N2) scaling at all N tested (up to Nelectron=864). While increasing the number of sampling points of the original orbital, the total QMC energy converges to the value calculated using plane waves at similar sampling point numbers for Lagrange, pp-splines and B-splines. At fixed sampling density, the three are of comparable speed. pp-splines use 8 memory words per sampling point, Lagrange use 5 and B-splines use 1. For B-splines at energy-converged sampling point number, the memory use can be as little as three times the storage requirements of plane waves. Due to comparable accuracy and speed with other polynomial approximants but smaller memory usage, B-splines are the best choice of polynomial basis for speeding up QMC calculations of periodic systems.
Key references 1. A. J. Williamson, Randolph Q. Hood and J.C. Grossman, Phys. Rev. Lett. 87, 246406 (2001) 2. D. Alfè and M. Gillan, Phys. Rev. B. 70, 161101 (2004)
[edit] Dissociation energy of the water dimer from Quantum Monte Carlo calculations
by Idoia G. de Gurtubay, R. J. Needs
Abstract We report a study of the electronic dissociation energy of the water dimer using quantum Monte Carlo (QMC) techniques. We have performed variational quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) calculations of the electronic ground state of the water monomer and dimer using all-electron and pseudopotential approaches. We have used Slater-Jastrow trial wave functions with B3LYP-like single-particle orbitals, into which we have incorporated backflow correlations. When backflow (BF) correlations are introduced, the total energy of the water monomer decreases by about 4-5 mHa, yielding a DMC energy of -76.42830(5) Ha, which is only 10 mHa above the experimental value. In our pseudopotential DMC calculations, we have compared the total energies of the water monomer and dimer obtained using the locality approximation with those from the variational scheme recently proposed by Casula [PRB 74, 161102(R) (2006)]. The time step errors in the Casula scheme are larger and the extrapolation of the energy to zero time step always lies above the result obtained with the locality approximation. However, the errors cancel when energy differences are taken, yielding electronic dissociation energies within error bars of each other. The dissociation energies obtained in our various all-electron and pseudopotential calculations range over only 0.44 kcal/mol and are in good agreement with experiment. Our calculations give dipole moments within 0.05 Debye of experiment for the water monomer and 0.02 Debye of experiment for the dimer.
[edit] On How Accurate DFT is for H Bonds in Small Water Clusters: Benchmarks Approaching the Complete Basis-Set Limit
by Biswajit Santra, Angelos Michaelides and Matthias Scheffler
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Abstract The ability of density-functional theory (DFT) exchange-correlation (xc) functionals to accurately describe H bonds between water molecules remains an important open question.This is true despite widespread application of DFT to the treatment of water in various environments, for example, liquid water, ice, adsorbed, and confined water. Here, we address this issue through a series of studies of small gas phase water clusters. Such clusters are important in their own right (implicated in phenomena such as acid rain formation and ice nucleation) and, moreover, are amenable to treatment with explicitly correlated quantum chemistry methods like Moller-Plesset perturbation theory (MP2). Using MP2 calculations as our reference, we have assessed the abilities of 16 DFT xc functionals to describe the energetics of the low energy isomeric structures of water dimers, trimers, tetramers, and pentamers. Errors from basis-set incompleteness have been minimized in both the MP2 reference data and the DFT calculations, thus, enabling a systematic evaluation of the true performance of the tested functionals. Among the functionals considered, the hybrid X3LYP and PBE0 functionals are the best: predicting H bonds strengths within 10 meV (0.3 kcal/mol) of MP2. Of the nonhybrid GGA functionals, mPWLYP and PBE1W perform the best. The popular BLYP and B3LYP functionals consistently underbind, and PBE and PW91 display rather variable performance with cluster size. Another interesting finding is non-negligible differences between PW91 and PBE. Extending the analysis to the question of how well DFT does at describing the energetic ordering of the low energy isomeric structures, we address the controversial question of the structure of the low energy isomer of the water hexamer. Our MP2 results, which are consistent with recent coupled cluster calculations [1], indicate that four isomers lie within 2 kcal/mol of each other and of these the "prism" is the lowest energy structure. However, none of the DFT functionals tested predict the prism to be the lowest energy structure. Instead they favor the more open "cyclic" or "book" structures. Analysis indicates that a poor description of the two-body (dimer) interactions within the hexamer, particularly for widely separated water molecules, is the main origin of this failure.
Key references [1] R. M. Olson and J. L. Bentz and R. A. Kendall and M. W. Schmidt and M. S. Gordon , J. Chem. Theory Comput. (in press)
[edit] Quantum Monte-Carlo studies of B, C and Al clusters
by Ching-Ming WEI, Cheng-Rong Hsing, Neil Drummond, Richard Needs
Abstract The study of the relative stability of clusters has been one of the major subjects in theoretical calculation. From the DFT result, it has been shown that the different exchange-correlation functionals may give discrepancy in the energy ordering. In this work we use quantum monte carlo (QMC) methods to study different cluster isomers: B_18, B_20, Al_13, Al_55 and C_20. Since QMC provides a highly accurate description in electron-electron correlation, the calculations will be helpful to examine the accuracy of various exchange-correlation functionals in medium-size clusters. In aluminum clusters, LDA, PW91 and PBE calculations result in an identical energy ordering with QMC, and thus indicates that, for non-magnetic metallic cluster, the DFT approach with different approximate functionals is reliable. In boron and carbon clusters, however, the different functionals yield distinct energy ordering. Our QMC results suggest that the PBE approximation can predict a more accurate energy ordering than LDA. In particular, we also investigate the energy difference due to the John-Teller effect in the C_20 isomers. The results indicate both LDA and PBE approximations fail to describe correctly the relative energy. The QMC calculations are done using the CASINO code.
[edit] A graph space approach to Quantum Monte Carlo
by James Spencer, George Booth, Alex J. W. Thom, and Ali Alavi
Abstract Slater determinants are an ideal basis for fermionic systems as they explicitly provide the required antisymmetric nature of the solution. A path integral approach in such a basis, however, remains impossible as the weight of each path depends on the product of the signs of the individual elements of the high-temperature density matrix, and this is in general very poorly conditioned. By working in a finite space, we can partially resum paths into objects called "graphs". We describe several methods to evaluate the energy based on this idea, namely stochastic sampling of graph space (restricted to small graphs - up to four vertices), or a direct evaluation of certain types of large graphs (called "stars"). We have applied both methods to a variety of molecular and periodic systems, including nitrogen, the neon dimer, ceria, magnesium oxide, silicon, diamond and lithium hydride.
